Tangent-intercept method

The tangent-intercept method can also be applied to the partial molar property change of mixing if is plotted vs. X2, the intercept gives (AV ,j,)i = Vi — Oi. 

Values of cb for various molalities were calculated from density data of Chemical Engineers Han ook (McGraw-Hill Book Company, Inc., New York, 1950). Partial molal volumes Vb were calculated from these density data by the method of tangent intercepts described by Lewis and Randall (loc, cit.f p. 39). Relative viscosities m/mb are available for NaCl solutions at 18°C. in International Critical Tables (Vol. V, p. 15). The accompanying tabulation summarizes the results of the various calculations, and Fig. 5.6 compares the calculated results with the experimental data as reported in International Critical Tables (Vol. V, p. 67). 

The partial enthalpies of Eq. (8.76) can be obtained in a number of ways, e.g., from activity-coefficient data [12] or from integral heats of solution by the method of tangent intercepts [22, 40]. Since, however, the equilibrium partial pressures of components A and C from these solutions are required over a range of temperatures in any event, the simplest procedure is that described earlier in connection with Fig. 8.2 and Eq. (8.3). For ideal solutions,, is the enthalpy of pure liquid J at /, for all Xj... 

If we have experimental data, or an equation that is believed to represent such experimental data for some extensive property as a function of concentration, we can compute the partial molar values by the method of tangent slopes or the method of tangent intercepts. In either case we could plot the data and make the geometric constructions. However, the mathematical procedure, which our computers can do for us is much more useful. 

Example 6.5 is clearly a longer and messier way to do by tangent slopes what we did more easily by tangent intercepts in Example 6.4. So why bother Because some functions are easier to do by tangent slopes, and this method appears in the historical literature, so you must understand it to understand that literature. Most often we use tangent intercepts and Eqs. 6.4 and 6.5... 

If we do that (Problem 6.9) we find analogs to most of the equations and procedures shown in this chapter, with partial molar properties replaced by those same properties divided by the molecular weight. Such properties have no common name perhaps they are best called partial mass properties. The most commonly seen application is with plots of enthalpy per unit mass vs. mass fraction, as shown for water and sulfuric acid in Figure 6.8. As shown in Problem 6.9, if we apply the method of tangent intercepts to this figure, the intercept values are the partial mass enthalpies. 

Example 6.7 Using Figure 6.8 and the method of tangent intercepts, estimate the partial molar enthalpies of both water... 

The method of tangent intercepts is shown in the main text, based on the partial molar equation, Eq. 6.5. It may also be shown purely geometrically as follows. Figure 6.13 is the same as Figure 6.6, but the point of tangency has been moved to move points b and d further apart, points c, d, and e have been added, and unnecessary text has been deleted. 

FIGURE 7.9 Finding the partial molar volume residual amethane by the method of tangent intercepts. The line is tangent to the curve at Jmethane = 0.784. Its intercept on the right hand axis is 0.6 ft / Ibmol. 

In principle, we should be able to begin with Eq. F.6, take the partial derivatives with respect to yi, and use the method of tangent intercepts to find fi/ytP. In practice, this is very difficult, because it is very difficult to write out the derivatives of z from a pressure-explicit EOS. Instead, we begin with Eq. 7.15, rewritten as... 

Figure 21.27. -plot (left) and otj-plot (right) of the same isotherm of a carbon aerogel. The f-plot results in an intercept dashed line) at zero layer thickness of an adsorbed volume of 150 cm (STP)/g and a slope corresponding to a monolayer volume of 35.9 cm (STP)/g (equal to a specific surface area of about 155 mVg). The dashed-dotted lines indicate the series of tangent used for the MP method. 

A general method to evaluate the partial molar quantities Xa and Xb in a binary mixtore is based on the variant of the method of intercepts described in Sec. 9.2.3. The molar mixing quantity AX(mix)/n is plotted versus xb, where AX(mix) is (X—nAX —n X ). On this plot, the tangent to the curve at the composition of interest has intercepts equal to Xa—X ... 

Figure 4.2 shows V, the mean molar volume of a solution of ethanol (component 1) and water (component 2), as a function of xi. Let x be a particular value of xi for which we desire the values of the partial molar quantities Vi and V2. To apply the method, we draw a tangent line to the curve at xi = Xj, as shown in the figure. The intercepts of this line at the edges of the figure give the values of the two partial molar quantities for the composition xi = Xj. A proof of the validity of this method is contained in Appendix D. 

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